Defeating Mecha-King Ghidora
Use King Ghidora. As soon as the match starts and it roars three times (one for each head), jump and press B (multi kick). After Mecha-King Ghidora freezes, attack the light on its chest the same way. When Mecha-King Ghidora attacks you, do not dodge (just block) unless it jumps. Once it finishes, repeat the steps above until you destroy each head. Note: Try not to use special attacks -- they are not very useful.
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Float
As Mothra, Rodan, or King Ghidorah, hold R to jump and float down more slowly. Note: This works especially well on the Moonbase level.

King Ghidorah: Keep Air Kicking
Play as King Ghidorah and jump by pressing R. Hold R then press B to kick. If you keep holding R and tapping B, you will stay floating and kicking.
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Monster Island from Godzilla: Destroy All Monsters Melee
Play as Godzilla or Mothra in story mode. Complete all eight levels to see the ending sequence. When you complete all eight stages, the background will resemble the Monster Island level from Godzilla: Destroy All Monsters Melee on the Gamecube.